Fluctuations in Polymers and Random Data Partial Differential Equations

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This project will investigate the effect of random noise on idealized models of physical systems. The addition of random noise sometimes simplifies the analysis of complex systems, but can also fundamentally modify their behavior and give rise to unexpected phenomena. Both situations apply to the two classes of systems studied in this project: so-called random polymer models, which describe noisy, growing interfaces in condensed matter, and nonlinear differential equations that are used to model quantum and classical waves. The project will also use mathematical statistical physics more generally as a motivation to train undergraduate and graduate students in probability theory, an essential tool in modern computer science and artificial intelligence.More specifically, the aim is to develop robust tools to estimate the fluctuations of random polymers and related diffusion models, with the ultimate goal of extending results obtained by exact algebraic methods to broader classes of models where exact methods do not apply. Concerning nonlinear wave and Schroedinger equations, the project will study the evolution of the distribution of solutions for given random initial data, with special emphasis on extremal situations like small nonlinearity or small dispersion.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.